Circulation of a stream function

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SUMMARY

The discussion centers on demonstrating that the circulation around a closed path about the origin for a 2D incompressible flow, defined by the stream function \(\Psi=(\Gamma/2\pi)\ln(r)\), equals \(\Gamma\) and is independent of \(r\). The key equation utilized is \(\Gamma=-\oint \mathbf{V} \cdot d\mathbf{s}\). The radial velocity component \(U_r\) is determined to be zero, leading to the conclusion that only the angular velocity component \(U_\theta\) contributes to the circulation calculation.

PREREQUISITES
  • Understanding of 2D incompressible flow dynamics
  • Familiarity with stream functions and their applications
  • Knowledge of circulation and its mathematical representation
  • Proficiency in vector calculus, particularly line integrals
NEXT STEPS
  • Study the derivation of circulation in fluid dynamics using line integrals
  • Explore the properties of stream functions in incompressible flows
  • Learn about the implications of angular velocity in fluid mechanics
  • Investigate the relationship between circulation and vorticity in fluid systems
USEFUL FOR

Students and professionals in fluid dynamics, particularly those studying incompressible flow and circulation concepts, will benefit from this discussion.

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Homework Statement


The stream function of a 2D incompressible flow is given by:

\Psi=(\Gamma/2pi)*ln(r)

Show that the circulation about a closed path about the origin is \Gamma and is independent of r.

Homework Equations



\Gamma=-\ointV dot ds

The Attempt at a Solution



So far I know that the radial velocity component Ur=(1/r)*d(Psi)/d(theta)=0 because Psi does not depend on theta.
This means that the angular velocity component is the only velocity component: Utheta=-d(Psi)/d(r)=-L/2pi(r)
After this point I am not exactly sure what to do.
 
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